Macrodispersion by diverging radial flows in randomly heterogeneous porous media

Radial flow takes place in a heterogeneous porous formation where the transmissivity T is modelled as a stationary random space function (RSF). The steady flow is driven by a given rate, and the mean velocity is radial. A pulse-like of a tracer is injected in the porous formation, and the thin plume spreads due to the fluctuations of the velocity which results a RSF as well. Transport is characterized by the mean front, and by the second spatial moment of the plume. We are primarily interested in tracer macrodispersion modelling. With the neglect of pore-scale dispersion, macrodispersion coefficients are computed at the second order of approximation, without neglecting the head-gradient fluctuations. Although transport is non-ergodic at the source, it is shown that ergodicity is achieved at small distances from the source. This is due to the fact that close to the source local velocities are quite large, and therefore solute particles become uncorrelated very soon. Under ergodic conditions, we compare macrodispersion mechanism in radial flows with that occurring in mean uniform flows. At short distances the spreading effect is highly enhanced by the large variability of the flow field, whereas at large distances transport exhibits a lesser dispersion due to the reduction of velocities. This supports the explanation provided by Indelman and Dagan (1999) to justify why the macrodispersivity is found smaller than that pertaining to mean uniform flows. The model is tested against a tracer transport experiment (Fernàndez-Garcia et al., 2004) by comparing the theoretical and experimental breakthrough curves. The accordance with real data, that is achieved without any fitting to concentration values, strengthens the capability of the proposed model to grasp the main features of such an experiment, the approximations as well as experimental uncertainties notwithstanding.     

Analytical model of solute transport by unsteady unsaturated gravitational infiltration

Penetration of reactive solute into a soil during a cycle of water infiltration and redistribution is investigated by deriving analytical closed form solutions for fluid flux, moisture content and contaminant concentration. The solution is developed for gravitational flow and advective transport and is applied to two scenarios of solute applications encountered in the applications: a finite pulse of solute dissolved in irrigation water and an instantaneous pulse broadcasted onto the soil surface. Through comparison to simulations of Richards' flow, capillary suction is shown to have contrasting effects on the upper and lower boundaries of the fluid pulse, speeding penetration of the wetting front and reducing the rate of drying. This leads to agreement between the analytical and numerical solutions for typical field and experimental conditions. The analytical solution is further incorporated into a stochastic column model of flow and transport to compute mean solute concentration in a heterogeneous field. An unusual phenomenon of plume contraction is observed at long times of solute propagation during the drying stage. The mean concentration profiles match those of the Monte-Carlo simulations for capillary length scales typical of sandy soils.     

Assessment of sorbent/water ratio effect on adsorption using dimensional analysis and batch experiments

Many factors affect adsorption phenomena in solid-liquid systems. One of the most important factors is the sorbent/water (S/W) ratio in the system. However, the effect of varying S/W ratios on the adsorption is still unclear. In this study, batch experiments were examined to observe the adsorption of four contaminants (copper, cadmium, Butachlor, and Deltamethrin) in six soils with texture ranging from silty clay to loamy sand and with different S/W ratios. Dimensional analysis was used to assess the relationship between adsorption phenomena and S/W ratio. We have assumed that the total amount of sorbate sorbed in soil is a function of the equilibrium concentration, the volume of sorbate solution, and the sorbent amount in the system. A power function (Freundlich-like) model was obtained from the dimensional analysis. It can describe precisely the adsorption phenomena of different sorbents and sorbates in the moisture regime of paddy soils. Therefore, proper adsorption parameters can be obtained by this power function model regardless of the solids effect, which can then be utilized to describe the fate of solute in soil using solute transport models.     

Sorption nonequilibrium during solute transport

The effects of pore-water velocity, solute hydrophobicity, and sorbent organic-carbon content on sorption nonequilibrium during solute transport were evaluated. Nonequilibrium transport was observed to increase with pore-water velocity, solute hydrophobicity, and sorbent organic-carbon content. Nonequilibrium transport of neutral organic compounds was not detected with low organic-carbon (TOC = 0.33 g kg⁻¹) aquifer material, but was detected on higher organic sorbents from the unsaturated zone (TOC = 2.6 g kg⁻¹) and the soil surface (TOC = 6.9 g kg⁻¹). For solute-sorbent combinations yielding retardation factors > 2, nonequilibrium during transport was observed. After experimentally accounting for slow solute diffusion in the aqueous phase and isotherm nonlinearity as potential contributors to nonequilibrium solute transport, sorption nonequilibrium was attributed to slow solute diffusion within the organic-carbon matrix.     

Application of continuous time random walk theory to nonequilibrium transport in soil

Continuous time random walk (CTRW) formulations have been demonstrated to provide a general and effective approach that quantifies the behavior of solute transport in heterogeneous media in field, laboratory, and numerical experiments. In this paper we first apply the CTRW approach to describe the sorbing solute transport in soils under chemical (or) and physical nonequilibrium conditions by curve-fitting. Results show that the theoretical solutions are in a good agreement with the experimental measurements. In case that CTRW parameters cannot be determined directly or easily, an alternative method is then proposed for estimating such parameters independently of the breakthrough curve data to be simulated. We conduct numerical experiments with artificial data sets generated by the HYDRUS-1D model for a wide range of pore water velocities (υ) and retardation factors (R) to investigate the relationship between CTRW parameters for a sorbing solute and these two quantities (υ, R) that can be directly measured in independent experiments. A series of best-fitting regression equations are then developed from the artificial data sets, which can be easily used as an estimation or prediction model to assess the transport of sorbing solutes under steady flow conditions through soil. Several literature data sets of pesticides are used to validate these relationships. The results show reasonable performance in most cases, thus indicating that our method could provide an alternative way to effectively predict sorbing solute transport in soils. While the regression relationships presented are obtained under certain flow and sorption conditions, the methodology of our study is general and may be extended to predict solute transport in soils under different flow and sorption conditions.     

Theoretical analysis of deviations from local equilibrium during sorbing solute transport through idealized stratified aquifers

This paper studies the spreading characteristics of reactive solute plumes in idealized stratified aquifers. The aquifer consists of two layers having different permeabilities with flow parallel to the stratification. The solute is assumed to adsorb onto the aquifer solids according to a first-order reversible kinetic rate law; the adsorption parameters are spatially uniform. We use the Aris moment method to examine analytically the time evolution of the lower-order spatial moments of the depth-averaged contaminant plume for an instantaneous input of mass. The results demonstrate that sorption kinetics cause the total dissolved mass and average velocity of the contaminant plume to decrease with increasing travel time. The plume variance is shown to depend upon three factors: intra-layer longitudinal dispersion, intra-layer kinetics, and vertical averaging. The results indicate that the relative importance of sorption kinetics diminishes as the permeability contrast between the layers increases. We present a simple criterion that can be used to assess the applicability of the local equilibrium assumption in idealized stratified systems.     

Probability density function of non-reactive solute concentration in heterogeneous porous formations

Available models of solute transport in heterogeneous formations lack in providing complete characterization of the predicted concentration. This is a serious drawback especially in risk analysis where confidence intervals and probability of exceeding threshold values are required. Our contribution to fill this gap of knowledge is a probability distribution model for the local concentration of conservative tracers migrating in heterogeneous aquifers. Our model accounts for dilution, mechanical mixing within the sampling volume and spreading due to formation heterogeneity. It is developed by modeling local concentration dynamics with an Ito Stochastic Differential Equation (SDE) that under the hypothesis of statistical stationarity leads to the Beta probability distribution function (pdf) for the solute concentration. This model shows large flexibility in capturing the smoothing effect of the sampling volume and the associated reduction of the probability of exceeding large concentrations. Furthermore, it is fully characterized by the first two moments of the solute concentration, and these are the same pieces of information required for standard geostatistical techniques employing Normal or Log-Normal distributions. Additionally, we show that in the absence of pore-scale dispersion and for point concentrations the pdf model converges to the binary distribution of [Dagan, G., 1982. Stochastic modeling of groundwater flow by unconditional and conditional probabilities, 2, The solute transport. Water Resour. Res. 18 (4), 835-848.], while it approaches the Normal distribution for sampling volumes much larger than the characteristic scale of the aquifer heterogeneity. Furthermore, we demonstrate that the same model with the spatial moments replacing the statistical moments can be applied to estimate the proportion of the plume volume where solute concentrations are above or below critical thresholds. Application of this model to point and vertically averaged bromide concentrations from the first Cape Cod tracer test and to a set of numerical simulations confirms the above findings and for the first time it shows the superiority of the Beta model to both Normal and Log-Normal models in interpreting field data. Furthermore, we show that assuming a-priori that local concentrations are normally or log-normally distributed may result in a severe underestimate of the probability of exceeding large concentrations.     

Analytical solution to transport in three-dimensional heterogeneous well capture zones

Solute transport is investigated in a heterogeneous aquifer for combined natural-gradient and well flows. The heterogeneity is associated with the spatially varying hydraulic conductivity K(x, y, z), which is modelled as a log-normal stationary-random function. As such, the conductivity distribution is characterized by four parameters: the arithmetic mean K(A), the variance sigma(Y)(2) (Y=lnK), the horizontal integral scale I of the axisymmetric log-conductivity autocorrelation and the anisotropy ratio e=I(v)/I (I(v) is the vertical integral scale). The well fully penetrates an aquifer of constant thickness B and has given constant discharge QB, while the background aquifer flow is driven by an uniform mean head-gradient, - J. Therefore, for a medium of homogeneous conductivity K(A), the steady-state capture zone has a width 2L=Q/(K(A)|J|) far from the well (herein the term capture zone is used to refer both to the zone from which water is captured by a pumping well and the zone that captures fluid from an injecting well). The main aim is to determine the mean concentration as a function of time in fluid recovered by a pumping well or in a control volume of the aquifer that captures fluid from an injecting well. Relatively simple solutions to these complex problems are achieved by adopting a few assumptions: a thick aquifer B>I(v) of large horizontal extent (so that boundary effects may be neglected), weak heterogeneity sigma(Y)(2)<1, a highly anisotropic formation e<0.2 and neglect of pore-scale dispersion. Transport is analyzed to the first-order in sigma(Y)(2) in terms of the travel time of particles moving from or towards the well along the steady streamlines within the capture zone. Travel-time mean and variance to any point are computed by two quadratures for an exponential log-conductivity two-point covariance. Spreading is reflected by the variance value, which increases with sigma(Y)(2) and I/L. For illustration, the procedure is applied to two particular cases. In the first one, a well continuously injects water at constant concentration. The mean concentration as function of time for different values of the controlling parameters sigma(Y)(2) and I/L is determined within control volumes surrounding the well or in piezometers. In the second case, a solute plume, initially occupying a finite volume Omega(0), is drawn towards a pumping well. The expected solute recovery by the well as a function of time is determined in terms of the previous controlling parameters as well as the location and extent of Omega(0). The methodology is tested against a full three-dimensional simulation of a multi-well forced-gradient flow field test ([Lemke, L., W.B. II, Abriola, L., Goovaerts, P., 2004. Matching solute breakthrough with deterministic and stochastic aquifer models. Ground Water 42], SGS simulations). Although the flow and transport conditions are more complex than the ones pertinent to capture zones in uniform background flow, it was found that after proper adaptation the methodology led to results for the breakthrough curve in good agreement with a full three-dimensional simulation of flow and transport.     

A two-region nonequilibrium model for solute transport in solution conduits in karstic aquifers

A two-region nonequilibrium model was used to calibrate initial solute-transport parameter estimates generated from tracer-breakthrough curves (TBCs) developed from tracer tests conducted in uni-axial solution conduits in karstic aquifers. Two-region nonequilibrium models account for partitioning of solute into mobile- and immobile-fluid regions to produce a more representative model fit to the strong tails associated with TBCs than do equilibrium models. The nonequilibrium model resulted in an increase in average flow velocities and a decrease in longitudinal dispersion coefficients over comparable estimates using an equilibrium model. Increases in velocity and decreases in dispersion were obtained at the expense of including parameters that describe solute partitioning and mass transfer rate for the mobile- and immobile-fluid regions. In addition, nonidentifiable sorption and mass transfer parameters for the immobile-fluid regions could only be described in terms of upper and lower bounds using readily determined identifiable ratios representing solute partitioning and system constraints based on known physical properties. The identifiable ratios and system constraints serve to minimize model nonuniqueness and renders the nonidentification problem trivial.     

Sorption of hydrophobic organic pollutants in saturated soil systems

Sorption equilibria and rates were characterized for a matrix of four aquifer sands and two slightly to moderately hydrophobic organic solutes (nitrobenzene and lindane), and the effects of sorption on the behavior of these solutes in saturated systems of the soils were determined. Experimental data were used to test and evaluate a variety of mathematical models for predicting contaminant fate and transport in groundwater systems.Observed equilibrium relationships between soil and solution phase solute concentrations were found to be described best by the nonlinear Freundlich isotherm model. It was further determined that the sorption process in the systems tested is rate controlled, requiring several days to approach equilibrium in completely mixed batch reactors. Subsequent modeling of solute transport in continuous flow soil column reactors was found to be most successful when rate-controlled models were used, the best results were obtained with a dual-resistance model incorporating the coupled mass transport steps of boundary-layer and intraparticle diffusion.     

Simulations of physical nonequilibrium solute transport models: Application to a large-scale field experiment

Equations expressing the spatial moments of solute concentration distributions simulated by various models, in terms of model parameters, have recently been presented. Using independently obtained parameter values, these equations are used to compare simulations of physical non-equilibrium models with spatial moment data collected in a large-scale natural gradient experiment on solute transport. The physical nonequilibrium models examined postulate the existence of layered zones of immobile water through which solute is transported by a diffusion mechanism. It is found that the qualitative aspects of the measured moment behavior are simulated by the physical nonequilibrium models if the independently obtained parameters are modified somewhat on the basis of reasonable corrective assumptions. It is further demonstrated that the physical nonequilibrium models, using parameter values obtained from spatial data, can qualitatively simulate temporal behavior at individual well points in this relatively homogeneous aquifer.     

Application of the method of temporal moments to interpret solute transport with sorption and degradation

In this note, we applied the temporal moment solutions of [Das and Kluitenberg, 1996. Soil Sci. Am. J. 60, 1724] for one-dimensional advective-dispersive solute transport with linear equilibrium sorption and first-order degradation for time pulse sources to analyse soil column experimental data. Unlike most other moment solutions, these solutions consider the interplay of degradation and sorption. This permits estimation of a first-order degradation rate constant using the zeroth moment of column breakthrough data, as well as estimation of the retardation factor or sorption distribution coefficient of a degrading solute using the first moment. The method of temporal moment (MOM) formulae was applied to analyse breakthrough data from a laboratory column study of atrazine, hexazinone and rhodamine WT transport in volcanic pumice sand, as well as experimental data from the literature. Transport and degradation parameters obtained using the MOM were compared to parameters obtained by fitting breakthrough data from an advective-dispersive transport model with equilibrium sorption and first-order degradation, using the nonlinear least-square curve-fitting program CXTFIT. The results derived from using the literature data were also compared with estimates reported in the literature using different equilibrium models. The good agreement suggests that the MOM could provide an additional useful means of parameter estimation for transport involving equilibrium sorption and first-order degradation. We found that the MOM fitted breakthrough curves with tailing better than curve fitting. However, the MOM analysis requires complete breakthrough curves and relatively frequent data collection to ensure the accuracy of the moments obtained from the breakthrough data.     

Grid lysimeter study of steady state chloride transport in two Spodosol types using TDR and wick samplers.

Solute transport in soils is affected by soil layering and soil-specific morphological properties. We studied solute transport in two sandy Spodosols: a dry Spodosol developed under oxidizing conditions of relatively deep groundwater and a wet Spodosol under periodically reducing conditions above a shallow groundwater table. The wet Spodosol is characterized by a diffuse and heterogeneous humus-B-horizon (i.e., Spodic horizon), whereas the dry Spodosol has a sharp Spodic horizon. Drainage fluxes were moderately variable with a coefficient of variation (CV) of 25% in the wet Spodosol and 17% in the dry Spodosol. Solute transport in 1-m-long and 0.8-m-diameter soil columns was investigated using spatial averages of solute concentrations measured by a network of 36 Time Domain Reflectometry (TDR) probes. In the dry Spodosol, solute transport evolves from stochastic-convective to convective-dispersive at a depth of 0.25 m, coinciding with the depth of the Spodic horizon. Chloride breakthrough at the bottom of the soil columns was adequately well predicted by a convection-dispersion model. In the wet Spodosol, solute transport was heterogeneous over the entire depth of the column. Chloride breakthrough at 1 m depth was predicted best using a stochastic-convective transport model. The TDR sampling volume of 36 probes was too small to capture the heterogeneous flow and concomitant transport in the wet Spodosol.     

Determination of bromacil transport as a function of water and carbon content in soils

This study was conducted to determine the significance of bromacil transport as a function of water and carbon content in soils and to explore the implications of neglecting sorption when making assessments of travel time of bromacil through the vadose zone. Equilibrium batch sorption tests were performed for loamy sand and sandy soil added with four different levels of powdered activated carbon (PAC) content (0, 0.01, 0.05, and 0.1%). Column experiments were also conducted at various water and carbon contents under steady-state flow conditions. The first set of column experiments was conducted in loamy sand containing 1.5% organic carbon under three different water contents (0.23, 0.32, and 0.41) to measure breakthrough curves (BTCs) of bromide and bromacil injected as a square pulse. In the second set of column experiments, BTCs of bromide and bromacil injected as a front were measured in saturated sandy columns at the four different PAC levels given above. Column breakthrough data were analyzed with both equilibrium and nonequilibrium (two-site) convection-dispersion equation (CDE) models to determine transport and sorption parameters under various water and carbon contents. Analysis with batch data indicated that neglect of the partition-related term in the calculation of solute velocity may lead to erroneous estimation of travel time of bromacil, i.e. an overestimation of the solute velocity by a factor of R. The column experiments showed that arrival time of the bromacil peak was larger than that of the bromide peak in soils, indicating that transport of bromacil was retarded relative to bromide in the observed conditions. Extent of bromacil retardation (R) increased with decreasing water content and increasing PAC content, supporting the importance of retardation in the estimation of travel time of bromacil even at small amounts of organic carbon for soils with lower water content.     

Assessing impacts of partial mass depletion in DNAPL source zones: II. Coupling source strength functions to plume evolution

Analytical solutions, describing the time-dependent DNAPL source-zone mass and contaminant discharge rate, derived previously in Part I [Falta, R.W., Rao, P.S., Basu, N., this issue. Assessing the impacts of partial mass depletion in DNAPL source zones: I. Analytical modeling of source strength functions and plume response. J. Contam. Hydrol.] are used as a flux-boundary condition in a semi-analytical contaminant transport model. These analytical solutions assume a power relationship between the flow-averaged source concentration, and the source DNAPL mass; the empirical exponent (gamma) is a function of the flow field heterogeneity, DNAPL architecture, and the correlation between them. The DNAPL source strength terms can account for partial source remediation, either at time zero, or at some later time after the DNAPL release. The transport model considers advection, retardation, three-dimensional dispersion, and sequential first-order decay/production of several species. A separate solution is used to compute the time-dependent mass of each contaminant in the plume. A series of examples using different values of gamma shows how the benefits of partial DNAPL source remediation can vary with site conditions. In general, when gamma>1, relatively large short-term reductions in the plume concentrations and mass occur, but the source longevity is not strongly affected. Conversely, when gamma<1, the short-term reductions in the plume concentrations and mass are smaller, but the source longevity can be greatly reduced. In either case, the source remediation effort is much more effective if it is undertaken at an early time, before much contaminant mass has entered the plume. If the remediation effort is significantly delayed, the leading parts of the plume are not affected by the source remediation, and additional control or remediation of the plume itself is required.     

Dual-domain solute transfer and transport processes: evaluation in batch and transport experiments

Soil macropore networks establish a dual-domain transport scenario in which water and solutes are preferentially channeled through soil macropores while slowly diffusing into and out of the bulk soil matrix. The influence of macropore networks on intra-ped solute diffusion and preferential transport in a soil typical of subsurface-drained croplands in the Midwestern United States was studied in batch- and column-scale experiments. In the batch diffusion studies with soil aggregates, the estimated diffusion radius (length) of the soil aggregates corresponded to the half-spacing of the aggregate fissures, suggesting that the intra-ped fissures reduced the diffusion impedance and preferentially allowed solutes to diffuse into the soil matrix. In the column-scale solute transport experiments, the average diffusion radius (estimated from HYDRUS-2D simulations and a first-order diffusive transfer term) was nearly double that of the batch-scale study. This increase may be attributed to a loss of pore continuity and a compounding of the small diffusion impedance through macropores at the larger scale. The column-scale solute transport experiments also suggest that two preferential networks exist in the soil. At and near soil saturation, a primary network of large macropores (possibly root channels and earthworm burrows) dominate advective transport, causing a high degree of physical and sorption nonequilibrium and simultaneous breakthrough of a nonreactive (bromide) and a reactive (alachlor) solute. As the saturation level decreases, the primary network drains, while transport through smaller macropores (possibly intra-ped features) continues, resulting in a reduced degree of nonequilibrium and separation in the breakthrough curves of bromide and alachlor.     

Extended geometric method: a simple approach to derive adsorption rate constants of Langmuir-Freundlich kinetics

A new and simple equation has been presented here for calculation of adsorption and desorption rate constants of Langmuir-Freundlich kinetic equation. The derivation of new equation is on the basis of extension and correction to the geometric method which has been presented by Kuan et al. [Kuan, W.-H., Lo, S.-L., Chang, C.M., Wang, M.K., 2000. A geometric approach to determine adsorption and desorption kinetic constants. Chemosphere 41, 1741-1747] for the kinetics of adsorption/desorption in aqueous solutions. The correction is to consider that the concentration of solute is not constant and changes as adsorption proceeds. The extension is that we applied Langmuir-Freundlich kinetic model instead of Langmuir kinetic model to consider the heterogeneity and therefore it is more applicable to the real systems. For solving Langmuir-Freundlich kinetic model, some geometric methods and also Taylor expansion were used and finally a simple and novel equation was derived (Eq. (20)) for calculation of adsorption rate constant. This new method was named "extended geometric method". The input data of the obtained equation can be simply derived from initial data of adsorption kinetics. Finally the adsorption of methyl orange onto granular activated carbon was carried out at dynamic and equilibrium conditions and the capabilities of extended geometric method were examined by the experimental data.     

Transport of polar and non-polar volatile compounds in polystyrene foam and oriented strand board

Transport of hexanal and styrene in polystyrene foam (PSF) and oriented strand board (OSB) was characterized. A microbalance was used to measure sorption/desorption kinetics and equilibrium data. While styrene transport in PSF can be described by Fickian diffusion with a symmetrical and reversible sorption/desorption process, hexanal transport in both PSF and OSB exhibited significant hysteresis, with desorption being much slower than sorption. A porous media diffusion model that assumes instantaneous local equilibrium governed by a nonlinear Freundlich isotherm was found to explain the hysteresis in hexanal transport. A new nonlinear sorption and porous diffusion emissions model was, therefore, developed and partially validated using independent chamber data. The results were also compared to the more conventional linear Fickian-diffusion emissions model. While the linear emissions model predicts styrene emissions from PSF with reasonable accuracy, it substantially underestimates the rate of hexanal emissions from OSB. Although further research and more rigorous validation is needed, the new nonlinear emissions model holds promise for predicting emissions of polar VOCs such as hexanal from porous building materials.     

Modelling the solute transport under nonequilibrium conditions on the basis of mass transfer equations

A solute transport model that describes nonequilibrium adsorption in soil/groundwater systems by mass transfer equations for film and intraparticle diffusion is presented. The model is useful in cases where breakthrough curve spreading cannot be explained by dispersion only. To evaluate its validity, the model was applied to several data sets from column experiments. The validity was also proved by a comparison with an analytical solution for the limiting case of predominating dispersion. Furthermore, a sensitivity analysis was performed to illustrate the influence of different process and sorption parameters (pore water velocity, intraparticle mass transfer coefficient, isotherm nonlinearity) on the shape of the calculated breakthrough curves. The application of the proposed model is discussed in comparison to the widely used dispersed flow/local equilibrium model, and a relationship between both models, which is based on a lumped parameter approach, is shown.     

Continuous time random walk model better describes the tailing of atrazine transport in soil

Contaminant transport in soils is complicated and involves some physical and chemical nonequilibrium processes. In this research, the soil column displacement experiments of Cl⁻ and atrazine under different flow velocities were carried out. The data sets of Cl⁻ transport in sandy loam fitted to the convection dispersion equation (CDE) and the two-region model (TRM) indicated that the effects of physical nonequilibrium process produced by immobile water on the breakthrough curves (BTCs) of Cl⁻ and atrazine transport through the repacking soil columns were negligible. The two-site model (TSM) and the continuous time random walk (CTRW) were also used to fit atrazine transport behavior at the flow rate of 19.86 cm h⁻¹. The CTRW convincingly captured the full evolution of atrazine BTC in the soil column, especially for the part of long tailing. However, the TSM failed to characterize the tailing of atrazine BTC in the soil column. The calculated fraction of equilibrium sorption sites, F, ranging from 0.78 to 0.80 for all flow rates suggested the contribution of nonequilibrium sorption sites to the asymmetry of atrazine BTCs. Furthermore, the data sets for the flow rates of 6.68 cm h⁻¹ and 32.81 cm h⁻¹ were predicted by the TSM and the CTRW. As to the flow rate of 6.68 cm h⁻¹, the CTRW predicted the entire BTC of atrazine transport better than the TSM did. For the flow rate of 32.81 cm h⁻¹, the CTRW characterized the late part of the tail better, while the TSM failed to predict the tailings of atrazine BTC.